Proof of breaking of self-organized criticality in a nonconservative abelian sandpile model

Annealed and quenched disorder in sand-pile models with local violation of conservation

In this paper we consider the Bak, Tang, and Wiesenfeld (BTW) sand-pile model with local violation of conservation through annealed and quenched disorder. We have observed that the probability distribution functions of avalanches have two distinct exponents, one of which is associated with the usual BTW model and another one which we propose to belong to a new fixed point; that is, a crossover from the original BTW fixed point to a new fixed point is observed. Through field theoretic calculations, we show that such a perturbation is relevant and takes the system to a new fixed point.

Criticality in conserved dynamical systems: experimental observation vs. exact properties

Conserved dynamical systems are generally considered to be critical. We study a class of critical routing models, equivalent to random maps, which can be solved rigorously in the thermodynamic limit. The information flow is conserved for these routing models and governed by cyclic attractors. We consider two classes of information flow, Markovian routing without memory and vertex routing involving a one-step routing memory. Investigating the respective cycle length distributions for complete graphs, we find log corrections to power-law scaling for the mean cycle length, as a function of the number of vertices, and a sub-polynomial growth for the overall number of cycles. When observing experimentally a real-world dynamical system one normally samples stochastically its phase space. The number and the length of the attractors are then weighted by the size of their respective basins of attraction. This situation is equivalent, for theory studies, to "on the fly" generation of the dynamical transition probabilities. For the case of vertex routing models, we find in this case power law scaling for the weighted average length of attractors, for both conserved routing models. These results show that the critical dynamical systems are generically not scale-invariant but may show power-law scaling when sampled stochastically. It is hence important to distinguish between intrinsic properties of a critical dynamical system and its behavior that one would observe when randomly probing its phase space.

Avalanche frontiers in the dissipative Abelian sandpile model and off-critical Schramm-Loewner evolution

Avalanche frontiers in Abelian sandpile model (ASM) are random simple curves whose continuum limit is known to be a Schramm-Loewner evolution with diffusivity parameter κ=2. In this paper we consider the dissipative ASM and study the statistics of the avalanche and wave frontiers for various rates of dissipation. We examine the scaling behavior of a number of functions, such as the correlation length, the exponent of distribution function of loop lengths, and the gyration radius defined for waves and avalanches. We find that they do scale with the rate of dissipation. Two significant length scales are observed. For length scales much smaller than the correlation length, these curves show properties close to the critical curves, and the corresponding diffusivity parameter is nearly the same as the critical limit. We interpret this as the ultraviolet limit where κ=2 corresponding to c=-2. For length scales much larger than the correlation length, we find that the avalanche frontiers tend to self-avoiding walk, and the corresponding driving function is proportional to the Brownian motion with the diffusivity parameter κ=8/3 corresponding to a field theory with c=0. We interpret this to be the infrared limit of the theory or at least a crossover.

Four height variables, boundary correlations, and dissipative defects in the Abelian sandpile model

We analyze the two-dimensional Abelian sandpile model, and demonstrate that the four height variables have different field identifications in the bulk, and along closed boundaries, but become identical, up to rescaling, along open boundaries. We consider two-point boundary correlations in detail, and discuss a number of complications that arise in the mapping from sandpile correlations to spanning tree correlations; the structure of our results suggests a conjecture that could greatly simplify future calculations. We find a number of three-point functions along closed boundaries, and propose closed boundary field identifications for the height variables. We analyze the effects of dissipative defect sites, at which the number of grains is not conserved, and show that dissipative defects along closed boundaries, and in the bulk, have no effect on any weakly allowed cluster variables, or on their correlations. Along open boundaries, we find a particularly simple field structure; we calculate all n-point correlations, for any combinations of height variables and dissipative defect sites, and find that all heights and defects are represented by the same field operator.

Conformal field theory correlations in the Abelian sandpile model

We calculate all multipoint correlation functions of all local bond modifications in the two-dimensional Abelian sandpile model, both at the critical point, and in the model with dissipation. The set of local bond modifications includes, as the most physically interesting case, all weakly allowed cluster variables. The correlation functions show that all local bond modifications have scaling dimension 2, and can be written as linear combinations of operators in the central charge -2 logarithmic conformal field theory, in agreement with a form conjectured earlier by Mahieu and Ruelle in Phys. Rev. E 64, 066130 (2001). We find closed form expressions for the coefficients of the operators, and describe methods that allow their rapid calculation. We determine the fields associated with adding or removing bonds, both in the bulk, and along open and closed boundaries; some bond defects have scaling dimension 2, while others have scaling dimension 4. We also determine the corrections to bulk probabilities for local bond modifications near open and closed boundaries.

Boundary conditions and defect lines in the Abelian sandpile model

We add a defect line of dissipation, or crack, to the Abelian sandpile model. We find that the defect line renormalizes to separate the two-dimensional plane into two half planes with open boundary conditions. We also show that varying the amount of dissipation at a boundary of the Abelian sandpile model does not affect the universality class of the boundary condition. We demonstrate that a universal coefficient associated with height probabilities near the defect can be used to classify boundary conditions.

Sandpile models and random walkers on finite lattices

Abelian sandpile models, both deterministic, such as the Bak, Tang, Wiesenfeld (BTW) model [P. Bak, C. Tang, and K. Wiesenfeld, Phys. Rev. Lett. 59, 381 (1987)] and stochastic, such as the Manna model [S.S. Manna, J. Phys. A 24, L363 (1991)] are studied on finite square lattices with open boundaries. The avalanche size distribution P(L)(n) is calculated for a range of system sizes, L. The first few moments of this distribution are evaluated numerically and their dependence on the system size is examined. The sandpile models are conservative in the sense that grains are conserved in the bulk and can leave the system only through the boundaries. It is shown that the conservation law provides an interesting connection between the sandpile models and random-walk models. Using this connection, it is shown that the average avalanche sizes <n>(L) for the BTW and Manna models are equal to each other, and both are equal to the average path length of a random walker starting from a random initial site on the same lattice of size L. This is in spite of the fact that the sandpile models with deterministic (BTW) and stochastic (Manna) toppling rules exhibit different critical exponents, indicating that they belong to different universality classes.

Finite-size effects of avalanche dynamics

We study the avalanche dynamics of a system of globally coupled threshold elements receiving random input. The model belongs to the same universality class as the random-neighbor version of the Olami-Feder-Christensen stick-slip model. A closed expression for avalanche size distributions is derived for arbitrary system sizes N using geometrical arguments in the system's configuration space. For finite systems, approximate power-law behavior is obtained in the nonconservative regime, whereas for N--> infinity, critical behavior with an exponent of -3/2 is found in the conservative case only. We compare these results to the avalanche properties found in networks of integrate-and-fire neurons, and relate the different dynamical regimes to the emergence of synchronization with and without oscillatory components.

Multiscale dynamics and robust critical scaling in a continuum current sheet model

We analyze the self-organized critical behavior of a continuum running avalanche model. We demonstrate that over local interaction scales, the model behavior is affected by low-dimensional chaotic dynamics that plays the role of the primary noise source. With the help of scale-free avalanches, the uncertainty associated with chaos is distributed over a variety of intermediate scales and thus gives rise to spatiotemporal fluctuations that are characterized by power-law distribution functions. We show that globally, the continuum model displays structurally stable critical scaling that can be observed in a finite region in the control parameter space. In this region, the system exhibits a power-law critical divergence of the integrated response function over a broad range of dissipation rates. The observed behavior involves a remarkably stable spatial configuration. We explain the robust features of the model by the adjustable dynamics of its global loading-unloading cycle, which allows maintaining the long-term stationary state without affecting the intrinsic avalanche dynamics.

Scaling fields in the two-dimensional Abelian sandpile model

We consider the unoriented two-dimensional Abelian sandpile model from a perspective based on two-dimensional (conformal) field theory. We compute lattice correlation functions for various cluster variables (at and off criticality), from which we infer the field-theoretic description in the scaling limit. We find perfect agreement with the predictions of a c=-2 conformal field theory and its massive perturbation, thereby providing direct evidence for conformal invariance and more generally for a description in terms of a local field theory. The question of the height 2 variable is also addressed, with, however, no definite conclusion yet.

Dissipative Abelian sandpiles and random walks

We show that the dissipative Abelian sandpile on a graph L can be related to a random walk on a graph that consists of L extended with a trapping site. From this relation it can be shown, using exact results and a scaling assumption, that the correlation length exponent nu of the dissipative sandpiles always equals 1/d(w), where d(w) is the fractal dimension of the random walker. This leads to a new understanding of the known result that nu=1/2 on any Euclidean lattice. Our result is, however, more general, and as an example we also present exact data for finite Sierpinski gaskets, which fully confirm our predictions.

Nonconservative abelian sandpile model with the bak-tang-wiesenfeld toppling rule

A nonconservative Abelian sandpile model with the Bah-Tang-Wiesenfeld toppling rule introduced by Tsuchiya and Katori [Phys. Rev. E 61, 1183 (2000)] is studied. Using a scaling analysis of the different energy scales involved in the model and numerical simulations it is shown that this model belongs to a universality class different from that of previous models considered in the literature.